Homogeneous bundles and the first eigenvalue of symmetric spaces

Universit`a di Milano–Bicocca Quaderni di Matematica

Homogeneous bundles and the first eigenvalue of symmetric spaces

Leonardo Biliotti, Alessandro Ghigi

Quaderno n. 4/2008 (arxiv:0709.2104v2)

Stampato nel mese di aprile 2008

presso il Dipartimento di Matematica e Applicazioni,

Universit`a degli Studi di Milano-Bicocca, via R. Cozzi 53, 20125 Milano, ITALIA

.

Disponibile in formato elettronico sul sito www.matapp.unimib.it.

Segreteria di redazione: Ada Osmetti - Giuseppina Cogliandro tel.: +39 02 6448 5755-5758 fax: +39 02 6448 5705

arXiv:0709.2104v2 [math.DG] 16 Mar 2008

Homogeneous bundles and the first eigenvalue of symmetric spaces

Leonardo Biliotti, Alessandro Ghigi

Abstract

In this note we prove the stability of the Gieseker point of an irre- ducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary K¨ahler metric on a compact Hermitian symmetric spaces of ABCD–type.

1 Introduction

Let X be a compact complex manifold and let E be a holomorphic vector bundle of rank r over X. The Gieseker point of E is the map

T

: Λ

H

(X, E) −→ H

(X, det E) (1) that sends an element s

∧ · · · ∧ s

∈ Λ

H

(X, E) to the section x 7→ s

(x) ∧

· · · ∧ s

(x) of H

(X, det E). This map was first considered by Gieseker in his work [10] in order to construct the moduli space of vector bundles on a projective manifold. He proved that for the set of Gieseker stable bundles E with fixed rank and Chern classes on a polarised (X, H) there is a uniform k

such that for k > k

, T

= T

is a stable vector (in the sense of geometric invariant theory) with respect to the action of SL H

(X, E(k))
on Hom Λ

H

(X, E(k)), H

(X, det E(k))

In this paper we consider the Gieseker point of homogeneous bundles . over rational homogeneous spaces. Such bundles are known to be Mumford- Takemoto stable [23], [24] and this implies they are Gieseker stable [16, p.191]. So we already know that after twisting with a sufficiently ample line bundle their Gieseker point is stable. The interest here is in the stability of T

itself, without allowing any twist. Our result is the following.

Theorem 1 Let E → X be an irreducible homogeneous vector bundle over

a rational homogeneous space X = G/P . If H

(E) 6= 0, then T

is stable.

We give two proofs of this result. The first is algebraic and uses a criterion of Luna for an orbit to be closed. This proof works over any algebraically closed field of characteristic zero. The second proof uses invariant metrics and relies on a result by Xiaowei Wang [25].

Our interest for this result is connected with a problem in K¨ahler ge- ometry. Consider a compact K¨ahler manifold X and fix a K¨ahler class a ∈ H

(X). Bourguignon, Li and Yau [4], gave an upper bound for the first eigenvalue of the Laplacian ∆

: C

(X) → C

(X) relative to any K¨ahler metric g whose K¨ahler form ω

lies in the class a. The bound depends on the numerical invariants (h

and degree) of a globally generated line bundle L over X. To get the best estimate one has to choose appropriately the bundle. As is shown in [4], if X = P

and L = O

(1) one gets the upper bound 2, which is optimal since it is achieved by the Fubini-Study metric.

In paper [2] Arezzo, Loi and the second author, generalised this result, by substituting a vector bundle E to the line bundle L. In this case one gets the same kind of estimate, but the vector bundle E must satisfy an additional condition, namely its Gieseker T

point must be stable. By this method one gets an upper bound for λ

on the complex Grassmannian [2].

Such a bound is optimal since it is achieved by the symmetric metric. It is important to notice that if one twists the bundle E by a positive line bundle H, the estimate gotten from the twisted bundle E(k) is very rough. In fact the estimate blows up as k → ∞ (see (12) below). So it is important to obtain some information on the stability of T

without twisting the bundle E.

The main motivation for the present work was to extend the estimate for λ

to other Hermitian symmetric spaces of the compact type using ap- propriate homogeneous bundles. We are able to prove the following.

Theorem 2 Let X be a compact irreducible Hermitian symmetric space of ABCD-type. Then

λ

(X, g) ≤ 2

for any K¨ahler metric g whose K¨ahler class ω

lies in 2π c

(X). This bound is attained by the symmetric metric.

It should be mentioned here that El Soufi and Ilias [7, Rmk. 1, p.96] have proved that the symmetric metric is a critical point (in suitable sense) for the functional λ

on the set of all Riemannian metrics with fixed volume.

Curiously in the two exceptional examples (E-type) the best estimate

gotten by this method is strictly larger than 2, which is λ

of the symmetric

metric.

Theorem 3 If X = E

/P (α

) (resp. X = E

/P (α

)) then λ

(X, g) ≤ 36

17 resp. λ

(X, g) ≤ 133 53 for any K¨ahler metric g whose K¨ahler class ω

lies in 2π c

(X).

It would be interesting to understand if this is a deficiency of the method, or if these symmetric spaces do in fact support metrics with λ

larger than 2.

Acknowledgements We wish to thank Prof. Gian Pietro Pirola and Prof. Andrea Loi for various interesting discussions. The second author is also grateful to Prof. Peter Heinzner for inviting him to Ruhr–Universit¨at Bochum during the preparation of this work. He also acknowledges partial support by MIUR COFIN 2005 “Spazi di moduli e teoria di Lie”.

2 Stability of the Gieseker point

Let X be a compact complex manifold and let E → X be a holomorphic vector bundle of rank r. Set

V = H

(X, E), V

= H

(X, det E) W = Hom(Λ

V, V

). (2) The algebraic group GL(V ) acts linearly on V hence on Λ

V . It therefore also acts on W. For a ∈ GL(V ) let Λ

a be the induced map on Λ

V . The action of GL(V ) on W is given by

a.T := T ◦ ( Λ

a)

. (3)

Consider the above action restricted to the subgroup SL(V ) ⊂ GL(V ). Ac- cording to the terminology of geometric invariant theory, a point T ∈ W is stable (for this restricted action) if the orbit of SL(V ) through T is closed in W and the stabiliser of T inside SL(V ) is finite. We denote by e S and S the stabilisers of T

in GL(V ) and SL(V ) respectively:

S = {a ∈ GL(V ) : a.T e

= T

} S = {a ∈ SL(V ) : a.T

= T

}. (4) For x ∈ X put

V

= {s ∈ V : s(x) = 0} V

= {t ∈ V

: t(x) = 0}.

The following two simple lemmata will be used in the (algebraic) proof of

Thm. 1.

Lemma 4 Let E be globally generated of rank r. Then a section s ∈ V belongs to V

if and only if for any choice of r − 1 sections s

, . . . , s

of E the section T

(s, s

, . . . , s

) of det E lies in V

.

The proof is immediate. Let Aut(E) be the group of holomorphic bundle automorphisms of E. If f ∈ Aut(E) and x ∈ X, the map f

: E

→ E

is a linear isomorphism. The function x 7→ det f

is holomorphic hence a (nonzero) constant and f 7→ det f is a character of Aut(E). For f ∈ Aut(E) and s ∈ H

(X, E) put

ε(f)(s) := f ◦ s. (5)

This defines a representation ε : Aut(E) → GL(V ).

Lemma 5 Let E be globally generated. Then ε({f ∈ Aut(E) : det f = 1}) = e S.

Proof. For f ∈ Aut(E) ε(f).T

(s

, . . . , s

)(x) = ε(f)

s

(x) ∧ · · · ∧ ε(f)

s

(x) =

= f

s

(x)

∧ · · · ∧ f

s

(x)

= det f

· s

(x) ∧ . . . ∧ s

(x) =

= det f

· T

(s

, . . . , s

)(x).

So ε(f).T

= det f

· T

. (6)

If det f = 1 then ε(f).T

= T

. This proves that ε({f ∈ Aut(E) : det f = 1}) ⊂ e S. Conversely, let a ∈ e S. We claim that a(V

) = V

for any x ∈ X.

Indeed let s ∈ V

. Then for any s

, . . . , s

∈ V

T

(as, s

, . . . , s

) = T

(as, aa

s

, . . . , aa

s

) =

= (a

. T

)(s, a

s

, . . . , a

s

) =

= T

(s, a

s

, . . . .a

s

)(x) = 0.

So as ∈ V

by Lemma 4, and indeed a(V

) = V

as claimed. We get therefore an induced isomorphism

f

: E

∼ = V/V

→ V/V

∼ = E

.

By construction f

(s(x)) = (a s)(x). Since E is globally generated this ensures that f is holomorphic so f ∈ Aut(E) and ε(f) = a. By (6) det f

T

= ε(f).T

= a.T

= T

. Since E is globally generated, T

6= 0 and it ·

follows that det f = 1. 

We recall two results that will be needed in the following.

Theorem 6 Let H be a reductive group and K ⊂ H a reductive subgroup.

Let X be an affine H-variety. If x ∈ X is a fixed point of K the orbit Hx is closed if and only if the orbit N

(K)x is closed.

This criterion is due to Luna [18, Cor. 1] and is based on the Slice Theorem.

For a complex analytic proof of the Slice Theorem see [12].

We recall that a rational homogeneous space is a projective variety X of the form G/P with G a simply connected complex semisimple Lie group and P a parabolic subgroup without simple factors. (See for example [1], [21], [3].) Such spaces are also called generalised flag manifolds. A homo- geneous vector bundle E over X is of the form E = G ×

U where U is a representation of P . If the representation is irreducible the vector bundle itself is called irreducible.

Theorem 7 ([23, Thm. 1]) Let E → X be an irreducible homogeneous vector bundle over a rational homogeneous space X. Then E is simple, i.e.

Aut(E) = C

· I

.

First proof of Thm. 1. Let X = G/P be as above. By Bott-Borel- Weil theorem (Thm. 13 below) the hypothesis H

(E) 6= 0 already ensures that E is globally generated, so both V and V

have positive dimension and T

6≡ 0. By the same theorem G acts irreducibly on both V and V

. Denote by ρ : G → GL(V ) and σ : G → GL(V

) these representations.

Since G is semisimple all the characters of G are trivial. In particular any representation of G on a vector space U has image contained in SL(U).

So in fact ρ : G → SL(V ) and σ : G → SL(V

). The Gieseker point is G-equivariant, that is

T

( Λ

ρ(g)(u)) = σ(g)(T (u)) u ∈ Λ

V. (7) Set H = SL(V ) × G and define a representation ̟ : H → GL(W) by

̟(a, g)T = σ(g) ◦ T ◦ Λ

a

.

Let τ : G → H be the morphism τ(g) = (ρ(g), g) and let K be the image of τ. K ⊂ H is a closed reductive subgroup and by (7) T

is a fixed point of K acting via ̟. We claim that the normaliser N

(K) is a finite extension of K. In fact, denote by Ad the conjugation on H. Given n ∈ N

(K) put

ϕ(n) = Ad(n)

: K → K. (8)

Let Aut(K) denote the group of automorphisms of K and Inn(K) the sub-

group of inner automorphisms. Then ϕ : N

(K) → Aut(K) is a morphism

of groups. Since K is semisimple Aut(K) is a finite extension of Inn(K) (see [13, p.423] or [20, Thm. 1 p.203]). Put N

= ϕ

(Inn(K)). Then N

⊳ N

(K) and

N

(K)/N

֒→ Aut(K)/ Inn(K).

Therefore N

(K) is a finite extension of N

and it is enough to prove that N

is a finite extension of K. Indeed if n ∈ N

there is some k ∈ K such that nk

n

= kk

k

for any k

∈ K. So k

n centralises K. If k

n = (a, g) (with a ∈ SL(V ) and g ∈ G) this means that for any g

∈ G we have

aρ(g

) = ρ(g

)a gg

= g

g.

The second formula says that g ∈ Z(G). The first formula says that a : V → V commutes with the representation ρ. Since this is irreducible Schur lemma implies that a = εI for some ε ∈ C

. But a ∈ SL(V ), so ε

= 1 where p = dim V . Denote by U

the group of p–roots of unity. Then k

n = (ε, g) ∈ U

× Z(G). This proves that the composition

U

× Z(G) → N

→ N

/K

is onto. Since Z(G) is finite, it follows that N

and N

(K) are finite ex- tensions of K. Now T

∈ W is a fixed point of K and N

(K) is a finite extension of K, so the orbit N

(K).T

is a finite set, hence it is closed.

Notice that both H and K are reductive. We can therefore apply Luna’s criterion (Thm. 6) to the effect that the orbit H.T

is closed. To finish we claim that H.T

= SL(V ).T

. Since the action (3) of SL(V ) and the restriction of ̟ to SL(V ) × {1} ⊂ H agree, the inclusion H.T

⊃ SL(V ).T

is obvious. For the other let h = (a, g) ∈ H. Then

̟(h)T

= σ(g) ◦ T

◦ Λ

a

= σ(g) ◦ T

◦ Λ

aρ(g)

· ρ(g)

=

= σ(g) ◦ T

◦ Λ

ρ(g)

◦ Λ

(ρ(g)a

) =

= T

◦ Λ

(ρ(g)a

) = ̟(aρ(g)

, 1)T

and aρ(g

) ∈ SL(V ). Therefore H.T

⊂ SL(V ).T

so the two orbits co- incide. This shows that the orbit of T

is closed. Let S and e S be the stabilisers defined as in (4). By Thm. 7, Aut(E) = C

· I

, therefore {f ∈ Aut(E) : det f = 1} is finite, which implies, by Lemma 5, that e S and

a fortiori S are finite. 

We remark that this proof works over any algebraically closed field of char-

acteristic zero.

We come now to the second proof of this result. Recall that if E is a glob- ally generated bundle on X and s = {s

, . . . , s

} is a basis of H

(X, E) there is an induced map ϕ

: X → G(r, N). Consider on G(r, N) the standard symmetric K¨ahler structure which coincides with the pull-back of the Fubini- Study metric via the Pl¨ucker embedding. Denote by µ : G(r, N) → su(N) the moment map for the standard action of SU(N) on G(r, N).

Theorem 8 ([25, Thm. 3.1]) Let (X

, ω) be a compact K¨ahler manifold and let E be a globally generated bundle on X. Then T

is stable if and only if there is a basis s of H

(X, E) such that

Z

X

µ ϕ

(x)

ω

(x) = 0. (9)

For the reader’s convenience we briefly sketch the proof.

Proof. Fix an arbitrary Hermitian metric h on E and consider on V the corresponding L

–scalar product. Let s be an orthonormal basis with respect to this product. On the line bundle det E consider the metric ϕ

h

where h

is the metric on O

(1). Consider on V

the corresponding L

– scalar product. Finally denote by h· , ·i

the Hermitian inner product on W,

|| · ||

being the corresponding norm. Since we have fixed a basis we may identify SL(V ) with SL(N, C). For g ∈ SL(N, C) set ν(g) = log ||g

.T

||

. We consider ν as a function on SL(N, C)/ SU(N). On this space Wang introduces another functional

L(g) :=

Z

M

X

I

||(g

T

)(s

)(x)||

ω

n! (x),

which is strictly convex on SL(N, C)/SU(N) [25, Lemma 3.5]. (Here s

= s

∧ · · · ∧ s

∈ Λ

V .) Critical points of L correspond to g ∈ SL(N) such that the basis {gs

, . . . , gs

} satisfies (9). For some constants C

, C

∈ R and C

, C

> 0 the inequalities

L ≥ C

ν + C

≥ C

L + C

(10) hold on SL(N, C)/SU(N). The first is proved by Wang [25, p.406]. The second is simply an application of Jensen inequality to the convex function

− log. If T

is stable, then ν is proper by the Kempf-Ness theorem [15].

Hence L is proper too, so admits a minimum and there is a basis s

such

that (9) is satisfied. On the other hand if there is such a basis, L has a

minimum and being strictly convex this means it is proper. By (10), ν is

proper as well and, again by Kempf-Ness theorem, this implies that T

is

stable. It should be noted that the identification of the moment map for a projective action with the differential of a convex functional is standard in analytic Geometric Invariant Theory [19, Ch. 8], [6, §6.5], [11]. 

Second proof of Theorem 1. Let K be a compact form of G = Aut(X).

By averaging on K we can find K-invariant metrics ω and h on X and E respectively. Let s be a basis of H

(X, E) that is orthonormal with respect to the L

-scalar product obtained using h and ω. By Bott-Borel-Weil theorem (Thm. 13 below) G and hence K act irreducibly on H

(X, E) ∼ = C

. Denote by σ : K → SU(N) this representation (recall that K is semisimple). Then µ ◦ ϕ

is K-equivariant and

B = Z

X

µ ϕ

(x)
ω

(x)

is a fixed point of ad(σ(K)) ⊂ GL(su(N)), that is σ(k)B = Bσ(k) for k ∈ K.

By Schur lemma this implies that B = λI, so B = 0 since B ∈ su(N). By

Thm. 8 the Gieseker point T

is stable. 

In order to clarify the meaning of the above result it might be good to notice that together with the numerical criteria of [10] it allows an easy proof of the Gieseker stability (see e.g. [16, p.189]) of irreducible homogeneous bundles. We sketch this argument, although a stronger result (Mumford- Takemoto stability) is well-known (see [21, p.65] and references therein).

.

Proposition 9 ([10, Prop. 2.3]) Let T ∈ W be a stable point. Let V

⊂ V be a subspace and let d a number 1 ≤ d < r. Assume that for any d+1 vectors v

, . . . , v

∈ V

, T (v

, . . . , v

, v

, · · · ) ≡ 0. Then dim V

<

(d/r) · dim V. If T is only semistable, then equality can hold.

In [10] there is a proof in the semistable case, which works as well in the stable case.

Corollary 10 Let E → X be an irreducible homogeneous vector bundle of rank r over a rational homogeneous space X = G/P . If H

(E) 6= 0, and F ⊂ E is a subsheaf of rank d, then h

(F ) < (d/r) · h

(E).

Fix now an irreducible homogeneous bundle E of rank r and let F ⊂ E be a

subsheaf of rank d, with 0 < d < r. Let H be any polarisation on X. Since

any line bundle is homogeneous, E(k) = E ⊗H

is homogeneous. By Serre Theorem there is a k

such that for k ≥ k

H

(X, F (k)) = H

(X, E(k)) = {0} i > 0

and both E(k) and F (k) are globally generated. By Thm. 1, T

is stable, so by the above corollary χ(X, F (k)) = h

(X, F (k)) < (d/r) · h

(X, E(k)) = (d/r) · χ(X, E(k)). This proves that any irreducible homogeneous bundle is Gieseker stable with respect to any polarisation.

3 The first eigenvalue of Hermitian symmetric spaces

Here we want to apply the previous stability result to a problem in spectral geometry. Let X be a projective manifold and L an ample line bundle on X. Let K(L) be the set of K¨ahler metrics g with K¨ahler form ω

lying in the class 2π c

(L). For g in K(L) let ∆

be the Laplacian on functions,

∆

f = −d

df = 2 g

∂

f

∂z

∂¯z

.

It is well-known that ∆

is a negative definite elliptic operator and has there- fore discrete spectrum: denote its eigenvalues by 0 > −λ

(g) > −λ

(g) >

· · · . The following result of Lichnerowicz relates λ

to K¨ahler-Einstein ge- ometry.

Theorem 11 ([9, Thm. 2.4.3, p.41]) If X is a Fano manifold and g

is a K¨ahler-Einstein metric, i.e. Ric(g

) = g

, then λ

(g

) = 2 if Aut(X) has positive dimension and λ

(g

) > 2 otherwise.

We are interested in upper estimates for λ

(g) of general metrics in the class K(L). Bourguignon, Li and Yau [4] first studied this problem and showed that the supremum

I(L) = sup

K(L)

λ

(g) (11)

is always finite. (This heavily depends on the restriction to K¨ahler metrics,

see [5].) They gave an explicit upper bound for I(L) in terms of numerical

invariants of a globally generated line bundle E. For (X, L) = (P

, O

(1))

they were able to show that I(L) = 2. The following criterion, due to

Arezzo, Loi and the second author, is an extension of Bourguignon, Li and

Yau’s theorem. It allows to attack this problem using holomorphic vector

bundles instead of just line bundles.

Theorem 12 ([2, Thm. 1.1]) Let (X, L) be a polarised manifold and E a holomorphic vector bundle of rank r over X. Assume that E is globally generated and nontrivial and put

J(E, L) := 2 dim

X · h

(E) hc

(E) ∪ c

(L)

, [X]i

r (h

(E) − r) hc

(L)

, [X]i . (12) If the Gieseker point T

is stable, then

I(L) ≤ J(E, L). (13)

The result of [2] is slightly more general since there is no projectivity as- sumption on X.

We want to apply this result to the case where X = G/P is a rational homogeneous space and E is homogeneous. In this case J(E, L) can be computed, at least in principle, in terms of Lie algebra data. To proceed we fix the following (standard) notation. (See e.g. [8, Ch. 1],[21], [3].) G is a simply connected complex semisimple Lie group, g = Lie G, h ⊂ g is a Cartan subalgebra, l = dim h is the rank of G, B is the Killing form of g, ∆ is the root system of (g, h), ∆

is a system of positive roots, ∆

=

−∆

, Π = {α

, . . . , α

} is the set of simple roots, ̟

, . . . , ̟

denote the fundamental weights. Λ = Z̟

⊕ · · · ⊕ Z̟

⊂ h

is the weight lattice of g relative to the Cartan subalgebra h. For α ∈ ∆ let H

∈ h be such that α(X) = B(X, H

). b is the standard negative Borel subalgebra:

b = h ⊕ M

α∈∆−

g

(14)

Parabolic subalgebras containing b are of the form

p(Σ) = b ⊕ M

α∈ span(Π−Σ)∩∆+

g

(15)

where Σ is some subset of Π. For example Σ = Π corresponds to b, Σ = ∅ to g and maximally parabolic subalgebras are of the form p(α

). The algebra p(Σ) admits a Levi decomposition p(Σ) = l(Σ) ⊕ u(Σ), where u(Σ) is the nilpotent radical and

l(Σ) = h ⊕ M

α∈ span(Π−Σ)∩∆

g

is the reductive part. This latter admits a further decomposition l(Σ) = z(Σ) ⊕ s(Σ), z(Σ) being the center and s(Σ) being semisimple. Moreover

z(Σ) = \

α∈Π−Σ

ker α ⊂ h

and s(Σ) = h

(Σ) ⊕ M

α∈ span(Π−Σ)∩∆

g

where h

(Σ) = span{H

: α ∈ Π − Σ} ⊂ h is a Cartan subalgebra for s(Σ) and h = z(Σ) ⊕ h

(Σ). We denote by B, P (Σ), L(Σ), U(Σ), Z(σ), S(Σ) the corresponding closed subgroups of G. Note that S(Σ) is simply connected.

One can describe p(Σ), P (Σ) and the homogeneous space G/P (Σ) by the Dynkin diagram of G with the nodes corresponding to roots in Σ crossed.

A weight λ = P

i

m

̟

∈ Λ is dominant for G or simply dominant if m

≥ 0 for any i. It is said to be dominant with respect to p(Σ) if m

≥ 0 for any index i such that α

6∈ Σ. By highest weight theory, the irreducible representations of G are parametrised by dominant weights, while irreducible representations of a parabolic subgroup P (Σ) are parametrised by weights that are dominant with respect to p(Σ). If λ is dominant we let W

denote the irreducible representation of G with highest weight λ. If λ is dominant for p(Σ) we let V

denote the irreducible representation of P (Σ) with highest weight λ. We let moreover E

denote the homogeneous vector bundle on X = G/P (Σ) defined by the representation V

, that is E

= G ×

V

. Theorem 13 (Bott-Borel-Weil) If λ ∈ Λ is dominant for G, then

H

(X, E

) = W

. Otherwise H

(X, E

) = {0}.

Bott’s version of the theorem is much more general, but this partial state- ment is enough for what follows. We also remark that if one chooses b to be the Borel subalgebra with positive instead of negative roots, which is cus- tomary for example in the usual picture of P

as the set of lines in C

, then one has to consider lowest weights instead of highest ones. This amounts to dualize both representations. With this choice the statement of the theorem becomes H

(X, E

) = W

. (The book [3] follows this convention.) Recall that the set of simple roots Π = {α

, . . . , α

} is a basis of Λ ⊗ Q.

For a weight λ ∈ Λ, let λ = P

i

ξ

(λ)α

be its expression in this basis. We say that the (rational) number ξ

(λ) is the coefficient of α

in λ. We denote by λ

the highest weight of the adjoint representation of G (that is the largest root).

Lemma 14 Let X = G/P (α

). The bundle E

associated to the funda-

mental weight ̟

is a very ample line bundle over X. Moreover Pic(X) ∼ =

H

(X, Z) = Z c

(E

). For any weight λ ∈ Λ that is dominant for P (α

) c

(E

) = dim V

ξ

(λ)

ξ

(̟

) c

(E

). (16) (For the proof see e.g. [23, §5.2], [21, p.56].)

In the following statement we summarise what we need of the structure theory of Hermitian symmetric spaces.

Theorem 15 An irreducible Hermitian (globally) symmetric space of the compact type is a rational homogeneous space. Moreover a rational homoge- neous space X = G/P is symmetric if and only if the representation of p on g/p induced from the adjoint representation of g is irreducible. The actual possibilities are explicitly listed in Table 1.

The characterisation in terms of irreducibility of g/p is due to Kobayashi and Nagano [17, Thm. A] (see also [3, p.26]).

Klein form Type

Grassmannian G

= SL(n)/P (α

) n ≥ 2 AIII Odd quadrics Q

= Spin(2n + 1)/P (α

) n ≥ 2 BI Even quadrics Q

= Spin(2n)/P (α

) n ≥ 3 DI Spinor variety X = Spin(2n)/P (α

) n ≥ 4 DIII Lagrangian

Grassmannian X = Sp(n, C)/P (α

) n ≥ 2 CI

X = E

/P (α

) EIII

X = E

/P (α

) EVII

Table 1: Irreducible Hermitian symmetric spaces of the compact type.

Proposition 16 Let X = G/P (α

) be a compact irreducible Hermitian symmetric space and let λ ∈ Λ be a nontrivial dominant weight. Then

J(E

, −K

) = 2 dim W

dim W

− dim V

· ξ

(λ)

ξ

(λ

) . (17)

Proof. The tangent bundle to X = G/P is the homogeneous bundle ob-

tained from the representation of P on g/p. For symmetric X this is ir-

reducible by Theorem 15, so Bott-Borel-Weil theorem and Lemma 14 ap-

ply. Since H

(X, T X) = g = W

(see [1, p.75, p.131]), g/p = V

and

T X = E

. Set m = dim X = dim V

. By Lemma 14 c

(−K

) = c

(T X) = m · ξ

(λ

)

ξ

(̟

) c

(E

), c

(E

) = dim V

· ξ

(λ)

ξ

(̟

) c

(E

), hc

(E

) ∪ c

(−K

)

, [X]i

hc

(−K

)

, [X]i = dim V

m · ξ

(λ) ξ

(λ

) .

The rank of E

is dim V

, while h

(X, E

) = dim W

by Bott-Borel-Weil theorem. Therefore

J(E

, −K

) = 2 m h

(E

)

r (h

(E

) − r) · hc

(E

) ∪ c

(−K

)

, [X]i hc

(−K

)

, [X]i =

= 2 m dim W

dim V

(dim W

− dim V

) · dim V

m · ξ

(λ) ξ

(λ

) =

= 2 dim W

dim W

− dim V

· ξ

(λ) ξ

(λ

) .

 We are now ready for the proof of theorems 2 and 3.

Proof of Theorem 2. Let X be a compact irreducible Hermitian sym- metric space. Denote by g

the symmetric (K¨ahler-Einstein) metric with K¨ahler form in 2πc

(X). We need to show that

I(−K

) = 2 = λ

(g

). (18) The second equality follows from Thm. 11. So I(−K

) ≥ 2 by definition (11). It is enough to prove that I(−K

) ≤ 2. For each space in the first five families in Table 3 we find a homogeneous bundle E

→ X such that J(E

, −K

) = 2. The result then follows applying Thm. 12. The relevant information regarding weights and degrees can be found for example in [14, p.66, p.69].

1. The case of the Grassmannians (type AIII) is settled by hand in [2, Thm. 1.3]. The vector bundle E is the dual of the universal subbundle. If we choose the Borel group as in (14) then E = E

.

2. For odd quadrics the Dynkin diagram is:

Q

= Spin(2n + 1)/P (α

) Type BI

− 1

n n

1 2 3

The largest root is λ

= ̟

. Put λ = ̟

. Then W

is the spin repre- sentation, while V

corresponds to the spin representation of the semisimple part S(α

) ∼ = Spin(2n − 1) of P (α

) . The bundle E

is the spinor bundle studied e.g. by Ottaviani [22]. Of course dim W

= 2

, dim V

= 2

. Finally ξ

(λ) = ξ

(̟

) = 1/2, ξ

(λ

) = ξ

(̟

) = 1, so

J(E

, −K

) = 2 dim W

dim W

− dim V

· ξ

(λ)

ξ

(λ

) = 2 · 2

2

− 2

· 1/2 1 = 2.

3. The situation is very similar for even quadrics. The Dynkin diagram is:

Q

= Spin(2n)/P (α

)

Type DI

− 1 n

1 2 3

n

The largest root is again λ

= ̟

. We take W

to be either one of the half-spin representation. (E

is one of the two spinor bundles on Q

, [22]). Say W

= S

. Then λ = ̟

and V

is the half-spin representation S

of S(α

) ∼ = Spin(2n−2). Now dim W

= 2

, V

= 2

, ξ

(̟

) = 1/2, ξ

(̟

) = 1, so again J(E

, −K

) = 2.

4. For the Lagrangian Grassmannian the Dynkin diagram is:

X = Sp(n, C)/P (α

) Type CI

− 1

n n

1 2 3

The highest weight of the adjoint representation is λ

= 2̟

. W

is the standard representation of Sp(n, C) on C

. The semisimple part of P (α

) is S(α

) = SL(n), so V

is the standard representation of SL(n) on C

. So choosing E = E

we get

J(E, −K

) = 2 · 2n

2n − n · ξ

(̟

) ξ

(2̟

) = 2.

5. For the Spinor varieties the Dynkin diagram is:

Spin(2n)/P (α

)

Type DIII

− 1 n

1 2 3

n

Take E = E

. W

is the standard representation of Spin(2n). The semisimple part of P (α

) is S(α

) = SL(n), so V

is the standard rep- resentation of SL(n) on C

. The largest root is λ

= ̟

, ξ

(̟

) = 1/2, ξ

(̟

) = 1. So

J(E, −K

) = 2 · 2n

2n − n · 1/2 1 = 2.

 Proof of Theorem 3. 1. For X = E

/P (α

) the Dynkin diagram (with Bourbaki numbering) is:

X = E

/P (α

) Type EIII

2

3 4

1 5 6

The largest root is λ

= ̟

. An easy computation gives J(E

, −K

)

= 36/17 and J(E

, −K

) = 78/31. If λ = P

i

a

̟

, then J(E

, −K

) ≥ 2 ξ

(λ)

ξ

(̟

) = 8

3 a

+ 2a

+ 10

3 a

+ 4a

+ 8 3 a

+ 4

3 a

. The right hand side is < 36/17 if and only if λ = ̟

or λ = ̟

. Therefore the best estimate is gotten with λ = ̟

.

2. For X = E

/P (α

) the Dynkin diagram (with Bourbaki numbering) is:

X = E

/P (α

) Type EVII

2

3 4

1 5 6 7

The largest root is λ

= ̟

. We have J(E

, −K

) = 133/53. If λ = P

i

a

̟

, then

J(E

, −K

) ≥ 2 ξ

(λ)

ξ

(̟

) = 2ξ

(λ) =

= 2a

+ 3a

+ 4a

+ 6a

+ 5a

+ 4a

+ 3a

.

The right hand side is < 133/53 if and only if λ = ̟

. Therefore the best

estimate is gotten with λ = ̟

. 

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Universit`a degli Studi di Parma,

E-mail: leonardo.biliotti@unipr.it Universit`a di Milano Bicocca,

E-mail: alessandro.ghigi@unimib.it